Download Synthesis and properties of single luminescent silicon quantum dots

ROYAL INSTITUTE
OF TECHNOLOGY
Synthesis and properties of single luminescent silicon quantum dots
ILYA SYCHUGOV
Doctoral Thesis
Stockholm, Sweden 2006
Royal Institute of Technology
School of Information and Communication Technology
Department of Microelectronics and Applied Physics
Akademisk avhandling som med tillstånd av Kungl. Tekniska högskolan framlägges till offentlig
granskning för avläggande av teknologie doktorsexamen i fysik fredag 19 januari 2006 kl 10:15 i
Electrum-3, Aula N2, Kungliga Tekniska Högskolan, Isafjordsgatan 28, Kista.
TRITA-ICT/MAP AVH Report 2007:1
ISSN 1653-7610
ISRN KTH/ICT-MAP/AVH-2007:1-SE
ISBN
91-7178-533-7
978-91-7178-533-6
© Ilya Sychugov, December 2006
Ilya Sychugov, “Synthesis and properties of single luminescent Si quantum dots”
Royal Institute of Technology, Laboratory of Materials and Semiconductor Physics, Stockholm, 2006.
Abstract
Silicon is an ubiquitous electronic material and the discovery of strong room temperature
luminescence from porous Si in 1990 raised hopes it may find a new lease of life in the
emerging field of optoelectronics. First, the luminescence was shown to be emitted from
nanostructures in a porous Si network. Later the same emission was seen from Si nanocrystals
and the concept of a Si quantum dot emerged. A number of different models have been
proposed for the origin of the light emission. Some involve interface states between a Si
nanocrystal and a surrounding shell, while others consider the effect of quantum confinement
in an indirect bandgap semiconductor.
In this work single Si nanocrystals were addressed to shed light on the mechanism of
luminescence. Nanocrystals were prepared using e-beam lithography with subsequent etching
and oxidation of silicon nanopillars. In particular, the non-uniform oxidation known as selflimiting oxidation was successfully used to form a single nanocrystal with characteristic size
of several nanometers inside a nanopillar. This preparation method allowed optical probing of
a single nanocrystal with far-field optics.
Results revealed sharp luminescence spectra at low temperatures with a linewidth of ~ 2 meV,
which is less than the corresponding thermal broadening. This property is a signature of
energy level discreteness, which is, in turn, a straightforward consequence of the quantum
confinement model. Another effect observed was a random on-off blinking, which is also
regarded as a hallmark of single fluorescent objects. This effect appeared to be dependent on
the excitation power density suggesting the involvement of Auger-assisted ionization in the
dynamics of nanocrystal luminescence. In addition, it was shown how a change in the
surrounding optical mode density affects the main parameters of luminescence from Si
nanocrystals, such as the radiative lifetime, the quantum efficiency and the total yield.
Finally, in order to clarify the influence of morphological properties, such as size or shape, of
a Si quantum dot on its luminescence, combined low-temperature photoluminescence and
transmission electron microscopy investigations were initiated. A method was developed
using focused ion beam preparation for such a joint characterization.
To conclude, the work presented in this thesis gives support to the quantum confinement
effect in explaining the light emission mechanism from nano-sized Si, as well as highlighting
the importance of both nanocrystal morphological structure and surrounding matrix optical
properties in the luminescence process.
Contents
List of publications…………………………………………………………………...1
List of symbols and acronyms…………………………………………………….....2
Acknowledgements…………………………………………………………………...3
1. Introduction……………………………………………………………….…..4
2. Quantum dots…………………………………………………………………6
2.1 Basic optical properties
2.1.1 Background………………………………………..………………………...6
2.1.2 Homogeneous linewidth of a quantum dot……..………………………..7
2.1.3 Blinking phenomenon……………………………..…………………...….11
2.2 Areas of application……………………………………………………….14
2.2.1 Quantum dots based lasers, LEDs and amplifiers..……………......... 14
2.2.2 Quantum dots as biological tags, units for quantum computing,...….16
3. Silicon as an active optical material…………………………..….…………18
3.1 Bulk material vs. nano-sized Si as a light emitter……………...………….18
3.2 Silicon nanocrystals
3.2.1 Porous silicon……………………………………………………….22
3.2.2 Si nanocrystals in SiO2 produced by implantation of Si atoms…….. 23
3.2.3 Other preparation methods………………………………………….24
3.3 Scope for development…………………………………...…..……….…...25
4 Experimental methods…………………………………………………….……30
4.1 E-beam lithography and post-lithography processing
4.1.1 Si nanopillars fabrication using positive resist……………………....30
4.1.2 Si nanopillars fabrication using negative resist………………….. …31
4.1.3 Preparation of a Si nanopillar for TEM imaging ………………. .…33
4.1.4 Formation of nanomesas in SiO2 doped with Si nanocrystals….. …35
4.1.5 Formation of SiO2 nanopillars doped with Si nanocrystals …... .....36
4.2 Photoluminescence measurements ………………..……………………37
5. Main results……………………………………………………………………40
5.1 Optical properties of a single silicon nanocrystal
5.1.1 Photoluminescence spectroscopy………………………………………..40
5.1.2 Blinking phenomenon……………………………… ………………...…43
5.2 Effect of optical mode density on luminescence parameters……...............44
6. Summary……………………………………………………………………….46
6.1 Comments on the appended papers ……………………………………….46
6.2 Conclusions and outlook……………………………………………………….48
Appendix A Blinking data analysis….…………………………………..……….……50
List of appended papers
__________________________________________________
[I] I. Sychugov, R. Juhasz, J. Valenta and J. Linnros “Narrow luminescence linewidth of a
silicon quantum dot”, Physical Review Letters 94, 087405, March, 2005;
[II] I. Sychugov, R. Juhasz, J. Linnros and J. Valenta “Luminescence blinking of a Si
quantum dot in a SiO2 shell”, Physical Review B 71, 115331, March, 2005;
[III] I. Sychugov, A. Galeckas, N. Elfstrom, A. R. Wilkinson, R. G. Elliman, and J. Linnros
“Effect of substrate proximity on luminescence yield from Si nanocrystals ”, Applied Physics
Letters, 89, 111124, 2006;
[IV] I. Sychugov, R. Juhasz, A. Galeckas, J. Valenta and J. Linnros “Single dot optical
spectroscopy of silicon nanocrystals: low-T measurements", Optical Materials 27, 5, 973-976,
February, 2005;
[V] I. Sychugov, R. Juhasz, J. Valenta, M. Zhang, P. Pirouz, J. Linnros “Light emission from
silicon nanocrystals: probing a single quantum dot”, Applied Surface Science 252, 15, 52495253, May, 2006;
[VI] I. Sychugov, J. Lu, N. Elfstrom, and J. Linnros “Structural and optical properties of a
single silicon quantum dot: towards combined PL and TEM characterization”, Journal of
Luminescence 121, 353 (2006).
1
List of symbols and acronyms
__________________________________________________
CCD – Charge-coupled device
CdSe – cadmium selenide
DOS – Density of states
E-beam – Electron beam (lithography)
EL – Electroluminescence
FIB – Focused ion beam
FWHM – Full-width at half-maximum
GaAs – gallium arsenide
HF – Hydrofluoric acid
HMDS – hexamethyldisilazane
HeCd – helium cadmium laser
LA – Longitudinal acoustic (phonon)
LDOS – Local density of optical (states) modes
LO – Longitudinal optical (phonon)
PL – Photoluminescence
RIE –Reactive ion etching
SEM – Scanning electron microscopy
Si - silicon
STM – Scanning tunneling microscopy
TA – Transverse acoustic (phonon)
TEM – Transmission electron microscopy
TO –Transverse optical (phonon)
VLSI – Very large scale integration
XPS – X-ray photoelectron spectroscopy
Z – Nucleus charge
aB – Exciton Bohr radius
Eg – Bandgap energy
kB – Boltzmann constant (1.38·1023 J/K)
µ-PL – microphotoluminescence
2
Acknowledgments
__________________________________________________
First, I wish to thank my supervisor during the first visit at Royal Institute of Technology,
Birger Emmoth, and department head, Ulf Karlsson, for their initial support at that time.
The foremost gratitude goes to the project supervisor, Jan Linnros, under whose guidance the
work was gently forwarded in the right direction. His supervising was neither distracting nor
ignoring all the way.
Next person to be mentioned is Jan Valenta, who founded a solid basis for this work. Indeed,
this thesis became possible to large extent due to his persistence in pursuing elusive photons!
I would like to thank Augustinas Galeckas for his thorough assistance in tons of practical
matters, being it hardware, software or just warehouse issues.
Robert Juhasz and Niklas Elfström are acknowledged for hours and hours spent in the clean
room processing our structures and never getting tired of it.
This work would be impossible to accomplish without our secretary, Marianne, who provided
endless support in matters, which most people unjustly consider as trivial.
I wish to thank all the department colleagues: Arturas, Martin, Leonardo, Antonio, Paulius,
Jens, Hanne, Maciej, Chris, Andrew Wilkinson for thesis proof-reading and many others.
John Österman for revealing a bit of traditional Sweden. Our magnificent “Real KTH” soccer
team, which rocketed from the fourth to the first division during these years, winning a winter
indoor cup on the way: Xavier, Anders, Martin and Martin, …
Finally, I would like to thank all colleagues at IMIT (ICT) for a nice working atmosphere.
Thanks a lot!
3
Chapter 1 Introduction_____________________________________________________________________
Chapter 1. Introduction
__________________________________________________
Silicon has emerged as a dominant material from the dawn of microelectronics back in 1950s. Today it still plays a fundamental role in the semiconductor industry. According to a recent
article (M. Ieong et al., Science 306, 2057; 2004) by 2016 inventive new device architectures
may well take silicon electronics comfortably into the regime where components are smaller
than 10 nm. On top of the related problems with existing crystal-growth methods, current
insulating materials and defect control, new physical effects are coming onto the scene. At
this range of sizes purely quantum effects, such as quantum confinement, become significant.
The quantum mechanical effects may impose insurmountable problems for future devices but
may, just as well, provide new pathways.
The quantum confinement effect, being a direct confirmation of elementary quantum
mechanics and the Schrödinger equation, has been widely investigated in direct bandgap
nanocrystals due to their straightforward optical applications. This research field goes back to
the beginning of the 1980-s, when the theoretical background was put forward. Nowadays,
some quantum dot related devices, such as lasers, can compete on equal terms with other
designs for occupying a niche in the market.
These two major semiconductor topics, silicon and nanocrystals, started merging together
with the prospect of application in optoelectronics in the beginning of the 1990-s, fostering a
strong interest in the research community. First in the form of nanoporous silicon, then as
nanocrystals (or, for amorphous phase, nanoclusters) embedded in a silicon dioxide matrix.
Although the on-chip laser appeared to be more difficult to reach than first anticipated, other
unforeseen applications, including ones in biology, gradually emerged.
The fabrication of silicon nanocrystals and the investigation of their optical properties on a
single-dot level in order to reveal the basic mechanism of luminescence was the main goal of
the present work. It is extremely difficult to control several nanometer size objects with
ordinary macroscopic techniques. Common methods produce compounds with a surface
density of nanocrystals of more than 1010 cm-2. This mass production is certainly desirable for
a number of applications. However, when basic physical properties are of interest, the
contribution from many nanoparticles to the signal and possible cross-talk between them
become an unwanted side-effect. Thus, a special preparation method must be used in order to
gain access to silicon nanocrystals on a single dot level. On the other hand, for optical
characterization, the main disadvantage of such nanocrystals is their low emission rate. In
practice this led to half an hour or even one hour long exposure times in order to get one
4
I. Sychugov
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
photoluminescence spectrum! In addition, the requirements on the mechanical and thermal
stability during the data acquisition were quite strict. However, when the experiments
eventually succeeded, the results appeared to be novel and of some interest to the community.
The thesis is organized as follows. First, the main concepts and basic properties of quantum
dots are introduced together with main areas for application and current trends (Chapter 2).
Second, the properties of silicon from the viewpoint of light extraction are discussed (Chapter
3). Methods and techniques used in this work are described in Chapter 4. An extended
summary of included papers is presented in Chapter 5 as well as the main results. A short
overview of the thesis and a brief summary constitute Chapter 6.
5
Chapter 2 Quantum dots___________________________________________________________________
Chapter 2. Quantum dots
__________________________________________________
2.1 Basic optical properties
2.1.1 Background
Bulk semiconductor materials have electrons with energy distributed in continuous bands,
separated by the energetically forbidden zone known as the bandgap. In the case of highpurity semiconductors at low temperature, the lower-lying valence band is filled and the other,
the conduction band, is empty. One can populate the conduction band with carriers by doping,
elevating temperature or providing the necessary energy, for example, by impinging light.
Reverse spontaneous transitions, from the conduction to valence bands, are accompanied by
energy emission as a propagating electromagnetic wave in order to obey the energy
conservation law. The light emission is quantized as photons and their energy directly related
to the bandgap energy.
In general, the existence of continuous bands can be considered as a result of an elementary
splitting of the degenerate levels of a single atom in the perturbing field of the other atoms
constituting a crystalline lattice. By reducing the number of perturbing atoms one can achieve
a structure where the continuous band of the bulk material transforms into a set of discrete
energy levels, similar to that of a single atom. This is schematically shown in Fig. 2.1, where a
Fig. 2.1 Density of states for various geometries of semiconductor materials: (a) 3-D bulk, (b)
quantum well, a 2-D structure, (c) quantum wire, a 1-D structure, and (d) quantum dot, a 0-D entity.
The dotted line in (a) is the thermal distribution of carriers [1].
6
I. Sychugov
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
continuous parabolic dependence of the density of states on energy in the bulk material
reduces into a number of discrete levels for a quantum box.
The characteristic size of a structure, when such a transition occurs, is defined by the exciton
Bohr radius. That is the typical size of a bound electron-hole pair in a semiconductor,
introduced in analogy with an electron Bohr radius of a hydrogen atom. This value is several
nanometers for most semiconductors. Thus these nanostructures possess unique optical and
electrical properties for solid state objects.
In addition to the discretization of energy levels when approaching nanometer range of sizes,
the value of the bandgap also changes. This effect can also be considered as a reduction in the
energy level splitting, bringing the valence and conduction bands further away from each
other. A well-known effect, similar to this one, is bandgap narrowing in bulk semiconductors,
where at lowering temperature the bandgap increases. The value of the bandgap increase can
be estimated for weak (a>aB) and strong (a<aB) confinement regimes from a quantum
mechanical “particle-in-a-box” problem [2]. The energy of an exciton in the weak
confinement regime is
E nml = E g −
Ry * h 2 χ 2ml
,
+
n2
2MR 2
(2.1)
where R is the QD radius, χ ml are roots of the spherical Bessel functions (m – number of the
e2
root, l – order of the function), Ry is the exciton Rydberg energy given by Ry =
, and
2εa B
*
*
P
P
M = me* + mh* . For the strong confinement regime:
E nl = E g +
h2
χ 2nl .
2µR 2
(2.2)
The graphical representation of these solutions is given in Fig. 2.2a, while in Fig. 2.2b the
numerical values of the bandgap are presented for a number of common semiconductors [2].
Fig. 2.2 (a) Quantum confinement effect in dimensionless units calculated using strong (a<aB, dashed
line) and weak (a>aB, dotted line) regimes approximations, (b) Corresponding bandgap energies for
various semiconductors as a function of quantum dot radius [2].
7
Chapter 2 Quantum dots___________________________________________________________________
Fig. 2.3 Ultra-narrow zero-phonon lines, observed at low-T from different single CdSe QDs [3].
2.1.2 Homogeneous linewidth of a quantum dot
The quantized levels of a quantum dot have been generally believed to be as sharp as a δfunction, reflecting its atomic character and size (Fig. 2.3). However, as a result of unique
dephasing mechanisms in quantum dots, the homogeneous width of a quantum dot is not
always as sharp as a δ-function, but is found to be of a finite width and is sometimes broader
than that of bulk crystals [1]. In this chapter the parameters that affect the homogeneous
linewidth of a quantum dot are discussed in more detail.
For simplicity, let us consider the case of a constant dephasing rate for a given excited state
(so-called Markovian relaxation). Here, the electron-phonon scattering is considered to be an
instantaneous process. Then, the corresponding line-shape is Lorentzian [2]:
I(ω) = A
Γ2
,
(ω − ω 0 ) 2 + Γ 2
(2.3)
where A – frequency independent coefficient, Г – phase relaxation rate (FWHM/2), ω0 resonant frequency.
The phase relaxation rate (dephasing rate) consists of a number of independent processes
characterized by the corresponding time constants:
Γ=
h
h
+∑ ,
2Tpop
i Ti
8
(2.4)
I. Sychugov
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
Fig. 2.4 Pathways for phase
relaxation of an exciton in a
QD. From left to right:
radiative e-h recombination
(exciton annihilation); phononassisted transition to the
ground state; acoustic phonon
mediated inelastic scattering;
acoustic phonon assisted elastic
scattering.
where Tpop – the pure population lifetime, Ti – the lifetime of the pure (adiabatic) dephasing
processes. The first term comes from real transitions of the relevant exciton state to the other
states (inelastic). The last term arises from the virtual processes that start from the relevant
state and, through some excursion in the intermediate states, return to the same initial state
(elastic). These pathways are schematically shown in Fig. 2.4.
Within the framework of the perturbation approach, for a two-level model consisting of the
ground |0〉 and the lowest excited state |1〉 of an exciton, the first, population term in (2.4):
Γpop = Γ0 + γ 01 ⋅ n (E 01 , T ) ,
(2.5)
h
, γ 01 – the rate of phonon-assisted transition
τ rad
|0〉 → |1〉, n (E 01 , T ) - Bose-Einstein distribution function, describing the number of available
phonons of suitable energy:
where Г0 – radiative annihilation rate: Γ0 =
n (E 01 , T ) =
1
.
exp(hω / kT) − 1
(2.6)
The second term in (2.5), contributing to the population broadening, depends on the
availability of the phonons with corresponding energy. Since LO-phonons energies can only
deviate weakly from the ELO value, the transition via LO-phonon(s) requires a strict matching
condition for a nanocrystal with a characteristic size “a”:
E 01 (a ) = m ⋅ E LO , m = 1, 2, … .
(2.7)
Precise quantum-mechanical calculations give a somewhat less strict condition (for the case
m=1):
E 01 (a ) = E LO ± δE, where δE ≈ 0.1 ⋅ E LO .
(2.8)
On the other hand, in nanocrystals even the acoustic phonon modes become discrete, which
reduces the probability of a multiparticle acoustic phonon assisted transition |0〉 → |1〉,
9
Chapter 2 Quantum dots___________________________________________________________________
providing that the value of E 01 >>ћωac, ћωac - acoustic phonon energy. It is important to note
that a trapped state, associated with the nanocrystal surface or defect with certain energy, may
act as the |1〉 state in the considerations above. Thus, the width of the transition can in
principle be dependent on the local environment.
Now, consider the second term in (2.4). The main mechanisms of dephasing are excitonphonon interaction, exciton-exciton interaction, scattering by defects or impurities, or
scattering by surfaces or interfaces [2]. Carrier-carrier interactions are absent under conditions
of low excitation when no more than one electron-hole pair within a crystallite is created. The
influence of defects can be neglected in perfect nanocrystals. Although at small sizes of
nanocrystals surface states may play a significant role, in most cases the second term can be
reduced to the exciton-phonon interaction only, which governs basic limits and temperature
dependence of the homogeneous linewidth.
According to [4], high-frequency optical modes interacting with the exciton manifests itself in
emission/absorption sidebands (phonon replicas), while low-energy acoustic modes cause an
energy fluctuation of the exciton level. Moreover, the most efficient coupling to the exciton
takes place for acoustic phonons of a certain wavevector and, correspondingly, of a certain
energy. According to [5], the homogeneous linewidth due to this fluctuation is proportional
to:
Γac ∝ D 2 ≡ χ 2 + 2χ 2 ⋅ n (hωac , T ) ,
(2.9)
where χ is the coupling constant (the matrix element of the electron-phonon interaction), and
n (hωac , T ) is Bose-Einstein statistics for the acoustic phonons with the most appropriate
energy. It was shown [4], that the low-energy phonons mostly contribute into this interaction.
Then for kT>> hωac the previous equation may be written as:
⎛ kT 1 ⎞
kT
Γac ∝ D 2 ≈ χ 2 + 2χ 2 ⋅ ⎜⎜
− ⎟⎟ = 2χ 2 ⋅
hωac
⎝ hωac 2 ⎠
(2.10).
To summarize, two different physical processes contribute to the homogeneous linewidth
broadening of a quantum dot. The dependence on temperature for both of them is determined
by the phonons which actually participate in the process. The pure dephasing process exhibits
a linear temperature dependence in the kT>> hωac regime, since mostly low-energy acoustical
phonons are involved. On the other hand, the population decay process takes place only when
the energy level structure of a nanocrystal allows phonon-assisted transitions for a given
exciton. In this case, the temperature dependence of the linewidth reflects Bose-Einstein
population statistics for the phonons with relevant energy.
As an example let us consider CdSe nanocrystals. The energy level splitting has been
calculated in detail, including the electron-hole exchange interaction and intrinsic hexagonal
lattice symmetry [5]. The calculations revealed the existence of so-called “dark” states, which
are optically passive. The splitting between the ground and the first optically active excited
state for the exciton was predicted to be in the range of 10-40 meV for a dot down to 2 nm in
size. The actual value of the splitting is largely dependent on the shape of the dot. With further
decreasing dot size the splitting grows significantly. Acoustic phonon spectra for a 2 nm CdSe
quantum dot calculated in [4] shows a set of discrete values from 1 to 12 meV. At the same
time the LO phonon energy 28 meV is nearly independent on the wavevector and nanocrystal
10
I. Sychugov
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
Fig. 2.5 Time-integrated four-wave mixing of InGaAs
QDs at different temperatures. Inset shows
corresponding FWHM estimated from the measured
dephasing of the quantum dot ground energy level
[10].
size. According to the calculations presented above, optical phonons can contribute to the
population decay rate only for a particular kind of CdSe nanocrystal, the so-called “phonon
bottleneck”.
For CdSe quantum dots the pure dephasing rate constants were calculated as a function of dot
size in [4]. The coefficient at the T-linear term of Гac from (2.10) for a 2 nm dot was estimated
to be ~50 µeV. The acoustic phonon energy corresponding to the maximum coupling with
excitons was found to be ~1 meV. Thus, a linear behavior of the linewidth temperature
dependence would be manifested at T > 30 K (for kT = 3 hωac the error for the Taylor
expansion of the Bose-Einstein statistics into linear terms is ~1%). Indeed, it was explicitly
shown [4] that the pure dephasing process can be used to interpret this commonly observed
linear dependence. The characteristic value for the emission linewidth of CdSe nanocrystals of
~20 meV at 80 K was detected experimentally and predicted theoretically on the basis of the
pure dephasing process [4, 7]. However, sometimes the best fit to experimental data can be
obtained by using simultaneously both processes responsible for the broadening [8].
When the temperature approaches the energy spacing between the ground and the first excited
state of a nanocrystal, a multiphonon assisted activation to the excited state becomes probable.
For example, for a 2 nm spherical CdSe dot the splitting value ~20 meV [5] corresponds
roughly to a temperature of T ≈ 200 K. Above this temperature the deviation from the linear
behavior was observed [4]. In the other extreme, for temperatures close to absolute zero, the
linewidth becomes ultranarrow, as would be expected for an atomic-like system when phonon
involvement becomes negligible. Values of 0.16-0.25 meV for CdSe dots at ~ 0 K were
deduced by the accumulated photon echo technique [9]. By single-dot spectroscopy, similar
widths were observed even at 10 K for some CdSe dots (Fig. 2.3, [3]). Four-wave mixing
experiments revealed even smaller values for InGaAs quantum dots (Fig. 2.5, [10]).
The observed difference in absolute values of the homogeneous linewidth at low temperatures
for nanocrystals prepared from various materials is a strong indication of the vital role played
by the surrounding matrix and the interface, which is unique for each case, in the light
emission mechanism of a quantum dot.
11
Chapter 2 Quantum dots___________________________________________________________________
2.1.3 Blinking phenomenon
The luminescence response of a quantum dot is not always constant in time. In addition to
possible irreversible photodegradation, there are reversible “dark” periods in otherwise stable
luminescence. Such sudden intermittence of light emission from quantum dots is normally
ascribed to random charging of the nanostructures, making non-radiative processes within
them dominant. It is believed that this charging effect results from interaction of the
nanocrystal with its close surrounding, namely due to a transfer of carriers to and from
quantum dots.
The most straightforward way of blinking analysis is based on a pre-estimated threshold
value, separating on and off states. By making a frequency count (statistical analysis) of the
recorded intensity values, one can distinguish between these two states in the occurrence
distribution and extract the value of the threshold. An example of the sporadic luminescence
behavior of a quantum dot is presented in Fig. 2.6, where an intensity trace is shown versus
time at different scales. Here the threshold can be intuitively placed at the level of 70-80
counts/10ms.
After the threshold value is established, the traces can be treated statistically, where the
number of “on” or “off” intervals of a certain length is plotted versus the length of the interval
(Fig. 2.7). Two principal kinds of distributions can be discerned: a single-exponential
distribution [11], and a power-law distribution [12]. The exponential distribution manifests a
purely random process with fixed transition rates:
⎛ t
p on ( t ) = A on ⋅ exp⎜⎜ −
⎝ τ on
⎞
⎛ t
⎟⎟ , p off ( t ) = A off ⋅ exp⎜⎜ −
⎠
⎝ τoff
⎞
⎟⎟
⎠
(2.11),
where Aon,off – amplitudes, τon and τoff – average times in on and off states.
The power-law dependence, on the other hand, represents a more deterministic one, featuring
certain physical restrictions (for example, dynamic transition rates):
Fig. 2.6 Luminescence intensity trace of a
CdSe quantum dot featuring intermittence
behavior [13].
12
I. Sychugov
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
Fig. 2.7 Two kinds of statistical distribution of the luminescence intermittence in quantum dots:
(left) exponential in a theoretical model [11] and (right) power-law observed for CdS QDs [12].
p on = A on ⋅ t α 0 , p off = A off ⋅ t α1
(2.12).
At the same time, these two can be present in one system, for example, one type of
distribution takes place for on-off transition statistics, while the other describes backward, offon transitions, and vice versa.
Another way to treat the recorded traces is using a normalized autocorrelation function of the
second order g(2) for the recorded intensity trace I(t):
g ( 2) (τ) =
< I ( t ) > ⋅ < I ( t + τ) >
< I( t ) > 2
(2.13),
and fitting the curves corresponding to ideal single-exponential or power-law distributions to
the obtained function, thus extracting main parameters of the blinking [12]. In this case there
is no need to set a threshold level, which can be quite ambiguous in certain cases.
The blinking effect and possible ways to suppress it have attracted strong interest in the
research community. Indeed, the blinking lowers the effective quantum efficiency for a QD
and thereby degrades the performance of any device built on ensembles of QDs. Recently, for
example, it was shown how this effect can be strongly suppressed by choosing an appropriate
buffer solution for CdSe nanocrystals [14]. In general, the blinking phenomenon highlights
the importance of the surrounding matrix and the interface role on the luminescence properties
of a quantum dot. Notably, many systems such as the self-organized InAs quantum dots show
no or negligible blinking. This may be ascribed to the smooth interphasing to surrounding
materials in the epitaxial growth process.
13
Chapter 2 Quantum dots___________________________________________________________________
2.2 Areas of application
2.2.1 Quantum dot based lasers
One of the main envisioned applications of quantum dots is involved with active optical
devices, such as lasers. In comparison to conventional atomic or solid state structures they
have a number of advantages, combining useful properties of these two (Fig. 2.8). An
example of laser design is shown in Fig. 2.9, where stacks of InGaAs quantum dots are
embedded into a multilayer structure. A laser based on quantum dots features tunability of
emission wavelength by the control of QDs size and composition.
In Fig. 2.10 the breakdown of the development of the threshold current density of
semiconductor lasers over last half century is shown. It is seen that devices built on quantum
dot ensembles exhibit a higher level of performance compared to their predecessors.
Fig. 2.8 Schematic representation of
the advantages in using QDs
compared to other media for lasing
applications (DOS – density of states,
Filling Factor – number of states per
volume unit)[15].
Fig. 2.9 Cross section of a
quantum dots based laser
with electrical injection of
carriers [16].
14
I. Sychugov
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
Fig. 2.10 Summary of the evolution of one of the key laser parameters: threshold current density at
room temperature. Quantum Dot based lasers represent a new generation of devices, following
Double HeteroStructure (DHS) and Quantum Well (QW) structures [17].
Another, more general, figure of merit is the beam quality factor M2. It is defined as a
standard in ISO 11140 as
1
4λ
1
⋅
=
2
π d σ0 ⋅ Θ σ
M
(2.14),
where dσ0 – size of beam-waist, Θσ – divergence angle, λ – lasing wavelength. Lower values
of this parameter correspond to better beam geometry. It is shown in Fig. 2.11 how this
parameter behaves under similar conditions in quantum well and quantum dot structures.
Again, the QDs configuration seems more preferable.
Fig. 2.11 Beam quality as a
function of the output power
for quantum well and quantum
dot configurations. Better values
are achievable using quantum
dots throughout a broader
operational range [18].
15
Chapter 2 Quantum dots___________________________________________________________________
2.2.2 Quantum dots as units for quantum computing, biological tags, etc
A two-state system with ground and excited states of a quantum dot can serve as a basis for
logical operations (Fig. 2.12). Here an excitation of an electron to the conduction band,
leaving behind a hole, could denote a |1〉-state. However, quantum mechanics allows the
system to be in a linear combination of the two states: |Ψ〉 = C0|0〉+C1|1〉. This can be called a
qubit (quantum bit). If the qubit state matches a measurement basis then a deterministic result
can be expected. However, dephasing (a random disturbing of the system by external forces)
imposes main limitations on the practical use of such systems. Long dephasing times are of
largest importance to enable at least one or a number of operations with these logical units.
That is why current prototypes work only at extremely low temperatures, where the dephasing
rate is suppressed due to low population of phonon modes. Scattering with phonons is the
main source of coherence loss and, hence, the capability of definite operations. On the other
hand, the advantage of using quantum dots is that they can be reliably loaded with individual
electrons and readily integrated into electronic devices.
The tunability of the emission wavelength of QDs together with their resistance to
photobleaching are widely used these days in biological applications. In Fig. 2.13 an example
of marking a biological object (a frog cell) with quantum dots is shown. QDs are used also as
tags, when attached to various molecules, thus tracking possible changes in complex systems.
Other applications of QDs include photodetectors, amplifiers, memories, single-photon light
sources and a number of others.
Fig. 2.12 Two-state configuration
of an exciton in a quantum dot,
which can be used for logical
operations: exciton qubit [19].
Fig. 2.13 Quantum dots injected into a frog cell are used as markers for monitoring cell development
[20]. (a) The QDs were injected in one cell at a stage where only 8 cells had been formed (upper left);
(b)-(e) shows further temporal evolution of the embryo with QDs marking the development of the
cell.
16
I. Sychugov
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
References
1. Y. Masumoto in Semiconductor Quantum Dots, edited by Y. Masumoto and T.
Takagahara (Springer-Verlag, Berlin, 2002), p. 325.
2. S. V. Gaponenko in Optical Properties of Semiconductor Nanocrystals (Cambridge
University Press, 1998), p. 111.
3. S. A. Empedocles, D. J. Norris and M.G. Bawendi, Phys. Rev. Lett. 77, 3873 (1996).
4. T. Takagahara, Phys. Rev. Lett. 71, 3577 (1993).
5. T. Takagahara, J. of Lumin. 70, 129 (1996).
6. A. L. Efros, M. Rozen, M. Kuno, M. Nirmal, D. J. Norris and M. G. Bawendi, Phys.
Rev. B 54, 4843 (1996).
7. S. V. Gaponenko, U. Woggon, A. Uhrig, W. Langbein and C. Klingshirn, J. of Lumin.
60, 302 (1994).
8. F. Gindele, K. Hild, W. Langbein and U. Woggon, Phys. Rev. B 60, R2157 (1999).
9. Y. Masumoto, M. Ikezawa, B. R. Hyun, K. Takemoto, M. Furuya, Phys. Status Solidi
B 224, 613 (2001).
10. P. Borri, W. Langbein, S. Schneider, U. Woggon, R.L. Sellin, D. Ouyang and D.
Bimberg, Phys. Rev. Lett. 87, 157401 (2001).
11. Al. L. Efros and M. Rosen, Phys. Rev. Lett. 78, 1110 (1997).
12. R. Verberk, A. M. van Oijen and M. Orrit, Phys. Rev. B 66, 233202 (2002).
13. M. Kuno, D. P. Fromm, H. F. Hamann, A. Gallagher and D. J. Nesbitt, J. Chem. Phys.
112, 3117 (2000).
14. S. Hohng and T. Ha, J. Am. Chem. Soc. 126, 1324 (2004).
15. D. Bimberg in “Semiconductor QDs” Summer School, 2004, Ascona, Switzerland.
16. M. Grundmann, “Semiconductor QDs” Summer School, 2004, Ascona, Switzerland.
17. N. N. Ledentsov, M. Grundmann, F. Heinrichsdorff, D. Bimberg, V. M. Ustinov, A. E.
Zhukov, M. V. Maximov, Z. I. Alferov and J. A. Lott, IEEE J. Sel. Top. Quantum El.
6, 439 (2000).
18. C. Ribbat, R. L. Selin, I. Kaiander, F. Hopfer, N. N. Ledentsov, D. Bimberg, A. R.
Kovsh, V. M. Ustinov, A. E. Zhukov, M. V. Maximov, Appl. Phys. Lett. 82, 952
(2003).
19. D. Gammon in “Semiconductor QDs” Summer School, 2004, Ascona, Switzerland.
20. B. Dubertret, P. Skourides, D. J. Norris, V. Noireaux, A. H. Brivanlou and A.
Libchaber, Science 298 (2002), 1759.
17
Chapter 3 Silicon as an active optical material_______________________________________________
Chapter 3. Silicon as an active optical material
__________________________________________________
3.1 Bulk material vs. nano-sized Si as light emitters
In general, there are three major recombination paths for excited carriers to recombine in bulk
Si: direct band-to-band transition, a transition with involvement of a phonon, and, finally, via
a defect state (which is normally a non-radiative process). The first process is usually
classified depending on the excess energy dissipation: a radiative recombination, where the
energy is transferred to the propagating wave, and Auger recombination, where the energy is
absorbed by another pre-excited free carrier. Due to the indirect band structure of Si (in Fig.
3.1(a) it is seen that minima of the conduction band and maxima of the valence band are
displaced in k-space) the direct band-to-band transition is of extremely low probability and
regarded as “forbidden”. On the other hand, when a phonon with a suitable momentum k is
involved, the k-conservation rule stands. Such phonon-assisted transitions actually dominate
in a measured PL spectrum of bulk Si (in Fig. 3.1b the peaks, corresponding to TO and TA-
Fig. 3.1 Si band structure (a) and low-temperature PL spectrum (b). Radiative recombination
involving at least one phonon (TO, TA, G is an optic mode at k=0, IV is a phonon mode
representing intervalley scattering of electrons) is much stronger than radiative recombination
involving no phonons (peak labeled 0) due to the indirect band structure of this material [1].
18
I. Sychugov
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
Fig. 3.2 (a) The most probable recombination path for carriers in bulk Si are non-radiative, via defect
states (shown as stars), due to the long radiative lifetime arising from indirect band structure of this
material; (b) at the same time in Si nanocrystals the localization of carriers promotes radiative bandto-band transitions [2].
phonon assisted processes, are prominent). It should be noted, though, that a three-particle
process of the electron-hole recombination with emission of a phonon has still relatively low
probability due to its many-body nature. Hence, the recombination becomes totally dominated
by non-radiative defect transitions. This prevails even in the high purity material as carrier
diffusion to defects or surface is quite efficient. Thus, bulk Si material is by no means a good
light emitting medium, with the fixed characteristic photon energy ~ 1.1 eV.
These properties change when silicon is nanostructured. The shielding of carriers from
escaping to defect centers enhances the probability of radiative recombination for the same
given concentration of defects (Fig. 3.2). Consider, for instance, Si nanoparticles of a few
nanometers size embedded in a SiO2 matrix. When optically pumped carriers are generated
inside these particles. In this case electron and hole wave functions are confined in all three
dimensions, which, in turn, makes possible otherwise forbidden in the bulk “no-phonon”
transitions. Although this kind of electron-hole recombination is sometimes referred to as
“quasidirect”, one should keep in mind that the concept of bands and, consequently, the
division of transitions into direct and indirect, is only valid under the assumption of an infinite
periodic lattice potential. Apparently, this assumption is too crude in such a case where the
exciton Bohr radius in a Si nanoparticle (aB = 4.3 nm) is comparable to the size of the particle
itself. As a result of the aforementioned effect of carrier confinement, light emission from Si
nanoparticles can be relatively strong. Using the coefficient for radiative recombination in
bulk Si at room temperature B ≈ 10-14 cm3s-1 [3] one can roughly estimate the radiative rate in
such nanostructures. If one assumes that a defect mediated recombination channel is
suppressed due to the spatial localization of carriers and Auger assisted non-radiative
transition is hindered for the same reason at low pumping regime of one exciton per
nanocrystal, then the bulk carrier density corresponding to one exciton in a ~ 3 nm quantum
dot is N ≈ 7x1019 cm-3. Finally, the radiative lifetime can be assessed as τRAD ≈ 1/(B·N) ≈ 1 µs.
This value is very close to the experimentally observed excitation lifetime values in Si
nanocrystals of the order of microseconds. In addition, due to the quantum confinement
19
Chapter 3 Silicon as an active optical material_______________________________________________
Fig. 3.3 The evolution of energy levels of a Si atom into continuous bands of bulk Si via hybridized
molecular orbitals of a Si cluster and discrete energy levels of a Si nanocrystal (compiled from
different sources, the basic concept is adopted from [4]). Upper row (from left to right): the energy
levels of a neutral Si atom calculated using Hartfree-Fock-Slater method [5]; four-electron sp3hybridized atoms forming molecular bond orbitals between neighbors – a set of bonding orbitals σ
and a set of antibonding orbitals σ* [4]; the Si/SiO2 interface energy diagram derived from XPS
measurements [6] (inset: hole and electron energy levels for Si nanocrystals with 5 and 8 nm in
diameter, as calculated in [7]); DOS for bulk Si as calculated in [8] with the vacuum level definition
from [9]. Lower row (from left to right): an artistic impression of a Si atom; one of a few possible Si
cluster configurations, as proposed in [10]; a TEM image of a Si nanocrystal in a SiO2 shell [11]; an
STM image of cleaved <111> 7X7 surface of bulk Si [12].
effect, the wavelength of light that is emitted from these nanoparticles can be tuned
continuously through the infrared and visible range with varying particle size.
In Fig. 3.3 the effect of size change on the electronic structure of a Si entity is shown as an
evolution of the discrete levels of an atom towards the continuous bands of the bulk material.
In this way, first, four-electron hybridized sp3 atoms create a set of bonding orbitals σ and a
set of antibonding orbitals σ*.
20
I. Sychugov
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
Fig. 3.4 Low temperature luminescence spectra of porous Si [17] and Si nanocrystals [18]. The
broad lines can be attributed to the inhomogeneous broadening.
Then, when the size of a Si structure increases, splitting of degenerate levels within a set of
orbitals leads to the formation of a number of discrete states, eventually becoming the wellknown conduction and valence bands of bulk Si with a bandgap energy Eg = 1.12 eV at room
temperature. It is interesting to note that atomic and molecular physics, quantum chemistry,
nanophysics and physics of semiconductors are all involved in such a representation,
highlighting the multidisciplinary nature of the current studies
In a closer look to Si nanocrystal electronic structure (inset in the upper panel in Fig. 3.3) one
can see that calculated energy levels of a nanocrystal are discrete and standing apart from one
another by a distance more than the thermal energy (kBT = 25 meV at RT) for small
nanocrystals. These calculations, performed using the [k·p] method, disregarded spin-orbit
interaction in Si [8]. However, since the first discovery of light emission from Si
nanostructures in 1990 (porous Si, [13]), no characteristic sharp lines in the PL spectra were
observed even at cryogenic temperatures, which was ascribed to inhomogeneous broadening.
At the same time, this fact boosted the appearance of other explanations of the light emission
mechanism from nano-sized Si, such as recombination via a stable oxygen-related defect
center created by annealing [14], involvement of surface states into the recombination process
[15], hot carrier ballistic transport towards the Si/SiO2 interface, enhancing the excitation of
oxide defect-related photoluminescence [16] and a number of others.
One of the main objectives of the current research project was, by probing properties of Si
nanostructures (nanocrystals, porous Si particles…) on a single-particle level, to clarify the
mechanism of their light emission. Such a single-dot PL spectroscopy method would be
extremely useful as it was the case in the studies of direct-bandgap nanocrystals [19]. The
much slower recombination rate in Si nanocrystals (excitation lifetime is 4-5 orders of
magnitude larger than in some direct bandgap QDs), however, was the main obstacle in the
way of such an experiment.
21
Chapter 3 Silicon as an active optical material_______________________________________________
3.2 Silicon nanocrystals
3.2.1 Porous Silicon
Historically, the first observations of strong room-temperature PL from silicon were
performed on porous silicon samples [13]. This sponge-like structure is usually obtained from
silicon by exposing it to HF solution under an anodic biasing condition. Depending on the
etching parameters, such as bias, HF concentration, hole density in silicon and temperature,
one can achieve a regime for pore formation (Fig. 3.5). The other regime is electropolishing,
when the whole surface of silicon recess uniformly.
It can be seen from Fig. 3.5 (right) that the characteristic size of silicon entities formed by
such a treatment may vary significantly. However, under certain circumstances, the size of
nanostructures can be tuned to yield structures in the several nanometers regime, where
luminescence in the visible range from silicon nanocrystals can be expected.
There were several attempts to utilize the luminescent properties of porous silicon in light
emitting diodes. In order to achieve efficient carrier injection into the porous silicon network
metal contacts can be placed in proximity to the porous silicon layer [22]. Another approach
is forming a vertical pn diode [23]. High quantum efficiencies were reported (external ~ 0.2
%), but degradation issues appeared to be an overwhelming obstacle to the practical
realization of such structures.
Recently a new application of these nanostructures was proposed. Due to the extremely large
surface area of porous Si the reaction of oxidation becomes energetically highly dense per
unit volume [24]. This leads to an explosive type of reaction under appropriate conditions.
This effect can be used in a controllable way. For example, porous Si can be filled with an
oxygen-rich substance and, when some threshold energy is transferred to this mixture, an
explosion may occur [25].
Fig. 3.5 Formation of porous silicon. (Left) SEM image illustrating the three typical etching regimes in
silicon: porous formation, transition regime with pillar-like structures, and electropolishing [20].
(Right) TEM micrograph of porous Si grains with characteristic size of several nanometers [21].
22
I. Sychugov
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
3.2.2 Si nanocrystals in SiO2 produced by implantation of Si atoms
An alternative method of nanocrystal manufacturing represents a bottom-up approach, where
atoms are precipitating from a supersaturated solution to form nanostructures of a desired
size. This is in contrast to the so called top-down concept, discussed previously on the
example of etching of bulk Si and porous silicon formation.
By creating an excess of Si atoms in an otherwise stoichiometric SiO2 matrix with subsequent
annealing one can trigger a precipitation process, where excess atoms diffuse to some
agglomerating centers [26]. One of the most straightforward ways to introduce such an excess
is by implantation of Si ions into silicon dioxide (Fig. 3.6). This method became widespread
due to its VLSI compatibility and good control of main parameters, such as the amount of
excess atoms (controlled by the implantation dose) and nanocrystal depth profile (by tuning
the implantation energy). This method was used in the present work also.
The energy of an ion beam can be changed from several keV, which corresponds to 10-20 nm
implantation depth, to several hundred keV, bringing the profile several hundred nanometers
below the surface. The effect of various implantation doses on obtained nanocrystal size and,
hence, on the luminescence spectra are shown in Fig. 3.6 (Right).
Recently, this technique was extended to implantation through a mask in order to form predefined luminescence patterns in silica [27] with a resolution of several hundred nanometers.
Fig. 3.6 (Left) Formation of silicon nanocrystals in a SiO2 shell by implantation of Si ions of 270
keV energy into silicon dioxide with subsequent annealing. (Right) Depending on the implantation
fluence [Si-/cm2] the mean size of nanocrystals may vary: with increasing fluence larger
nanoparticles are created, shifting the luminescence peak closer to the value of bulk Si ~ 1.1 µm.
23
Chapter 3 Silicon as an active optical material_______________________________________________
3.2.3 Other preparation methods
There are other ways to create non-stoichiometric films with Si, annealing of which can yield
nanocrystals. One of them is co-deposition of materials in, for example, an rf magnetron
discharge (Fig. 3.7, left). Here the sputtering targets can be chosen arbitrary so that the
chemical composition of the film may vary. Usually, the plasma is struck in an inert gas like
Ar and by means of physical sputtering elements from the targets can reach the substrate and
form composite films. By changing the volume ratio of targets it is possible to control the
relative chemical composition of the obtained sample. The advantage of this method is the
large choice of initial materials, which can be used for manufacturing of nanocrystals in
various matrices. For instance, Ge quantum dots embedded into silicon dioxide can be formed
by this technique [29].
Silane gas (SiH4) decomposition in various plasma reactors, like PECVD [30] or others [31]
with subsequent accumulation of the products on a substrate is another widely used method
for nanocrystal formation. Here, initial mixture of gases can be modulated during the process
in order to control the excess of Si atoms in the film [30]. In Fig. 3.7, right, the scheme of a
reactor [31], where a non-thermal plasma is created with hot electrons (2-5 eV) and cold, RT
atoms and ions is shown. This configuration enables formation of large quantities of
nanocrystals on the substrate on a very short time scale with suppression of excessive
agglomeration due to negative charging of the particles.
Fig. 3.7 (Left) Magnetron rf sputtering device, similar to the one used in the pioneering work [28].
(Right) Silane decomposition unit, where a gas discharge is used to form silicon nanocrystals [30].
24
I. Sychugov
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
3.3 Scope for development
Recent trends in the research of optical properties of silicon nanocrystals include attempts
aimed at improving the luminescent parameters, such as lifetime, using, for example, surface
plasmons [32] or photonic crystals [33]. Small metallic structures can serve as nano-antennas
for enhancing the spontaneous radiative recombination rate in silicon nanocrystals. The
oscillations of free electrons (plasmons), which can easily react on the incoming excitation,
perturb local near-fields, thus affecting luminescence parameters of nanocrystals brought in
proximity to these metallic objects (Fig. 3.8). The other approach deals with changing of the
optical mode density around nanocrystals by placing them in a photonic crystal. The physical
foundation of this method is the fact that spontaneous recombination rate W(r) depends on the
number of available final states, according to Fermi’s Golden Rule:
2 ⎛ ρ(ω, r ) ⎞
2
⎟⎟ ≡ M ⋅ LDOS(r)
W (r ) ~ M ⋅ ⎜⎜
⎝ ε( r ) ⎠
(3.1),
where |M| – matrix element of the interaction and LDOS (r) – local density of optical modes.
Photonic crystals are periodic structures with lattice spacing in the order of the designed
wavelength, which can be treated in a similar way as solid-state crystals. An analogy is that
electron behavior in the energy bands of bulk materials is similar here to the photon
propagation in such a structure. Thus, the pattern of light propagation can be changed as well
as the local light velocity enabling such features as negative index of refraction or the slowing
of light. Hence, luminescence can be collected more efficiently at some points or in some
directions. However, these structures are capable of not only acting passively, similar to
diffraction gratings, but can also affect the excitation lifetime. It was shown that a photonic
crystal with a bandgap (span of wavelengths forbidden for propagation, as an energy bandgap
for electrons in a semiconductor) inhibits radiation for in-plane waves, increasing the
excitation lifetime [34]. In a continuation to this approach one can benefit from the possible
coupling of a quantum dot to cavity modes in the photonic crystal [35].
Fig. 3.8 Enhancement of luminescence
from silicon nanocrystals caused by the
proximity of metallic islands acting as
nano-antennas: for the following
parameters p=400 nm, ∆ = 13nm, and
d=165, 185, 190, 230, 260 and 320 nm
(right, from top to bottom), the resonant
wavelength of the PL enhancement shifts
accordingly [32].
25
Chapter 3 Silicon as an active optical material_______________________________________________
Fig. 3.9 Silicon QDs as sensitizers for erbium: high absorption cross-section of Si nanocrystals, their
larger bandgap than corresponding interlevel spacing in Er3+, and lower energy backtransfer rate due
to spatial decoupling are the main advantages of such a mechanism of erbium excitation [40].
On the other hand, long excitation lifetimes in silicon nanocrystals can be advantageous for
some applications. In this respect one of the most promising is sensitization. In general,
nanocrystals can be excited more efficiently than atoms, which typically require strict
resonant excitation conditions. Then, via a near-field energy transfer mechanism, such as the
Förster transfer, excitation can be non-radiatively passed to desired atomic species. Usually,
these acceptors are erbium [36] or oxygen atoms [37]. The former is of interest in the field of
telecommunication due to the match of certain interlevel transitions in Er+3 with the
absorption minima in standard silica optical fibers at 1.54 µm (Fig. 3.9). The latter
sensitization mechanism can create the chemically active singlet oxygen state, normally
achieved only at elevated temperatures, thus promoting controlled “cold” reaction of
oxidation. This can be of use even in cancer treatment methods, where nanocrystals are first
localized in the tumor area, then irradiated with coherent light [38]. When the energy is
transferred to surrounding oxygen, it becomes aggressive towards cells selectively in this part
of the body.
Another important aspect is the control and correct assessment of luminescence quantum
efficiency (the relative number of radiative transitions). Typically the reported values are in
the order of tens of percents, which make silicon nanocrystals comparable to their direct
bandgap counterparts from this viewpoint. There are reports on even higher values for a
fraction of nanocrystals [39]. Up to date the consensus is that the quantum efficiency of
luminescent nanocrystals is high; however the quantum yield (the relative number of optically
active nanocrystals) is typically low. That suggests the existence of two kinds of defect states
in nanocrystals: one that completely quenches luminescence and the other with characteristic
time comparable to the radiative recombination time. This leads to strong absorption and poor
overall conversion of energy into useful propagating electromagnetic radiation for these
nanostructures.
26
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Synthesis and Properties of Single Luminescent Silicon Quantum Dots
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Synthesis and Properties of Single Luminescent Silicon Quantum Dots
29
Chapter 4 Experimental methods___________________________________________________________
Chapter 4 Experimental methods
__________________________________________________
Our distinctive approach in fabrication of single Si quantum dots for optical characterization
involved manufacturing of silicon nanopillars with subsequent oxidation. The pillars were
arranged in a pre-defined array using e-beam lithography. The separation between nanopillars
was ~ 1 µm, which was chosen to overcome the diffraction limit of light, thus enabling farfield optical characterization. The subsequent self-limiting oxidation of silicon nanostructures
at 900 oC, below the temperature of silicon dioxide flow, led in some cases to the formation of
a silicon nanocrystal inside nanopillars. In this temperature range the concave outer shape of a
silicon nanostructure undergoes faster oxidation, while the oxidation rate of a convex surface
is somewhat lower. This effect was reported in application to silicon nanopillars in the
beginning of 1990-s [1]. Below, experimental procedures of nanostructure fabrication used in
this work are described.
4.1 Electron beam lithography and post-lithography processing
Electron beam lithography is a tool for nanostructure manufacturing but mainly for research
purposes due to its low throughput rate. The electron beam hardens (negative) or softens
(positive) a resist layer on a chip where exposed (Fig. 4.1). Subsequent chemical treatment
(development) selectively targets non-exposed or exposed areas. Thus, created patterns can be
transferred down to the chip by, for example, reactive ion etching in a plasma reactor.
4.1.1 Si nanopillars fabrication using positive resist
The following experimental procedure was used to fabricate silicon nanopillars containing a
quantum dot, using positive resist as a mask for etching (Fig. 4.2):
-
thermal growth of a ~ 20 nm dioxide layer on top of a (001) n-doped 20-40 Ω·cm Si
substrate;
-
coating of the substrate with ~ 60 nm of positive resist ZEP 520A diluted in anisole
with 1:2 ratio;
30
I. Sychugov
-
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
e-beam lithography to open dots in the resist layer with diameter defined by the prearranged design, typically 100-200 nm;
-
deposition of a metal film (~ 15 nm of NiCr) and lift-off in ultrasound to invert the
pattern;
-
wet etching of the dioxide layer in HF using metallic dots as a mask;
-
wet removal of the NiCr mask;
-
RIE of the silicon substrate using dioxide islands as a mask to form silicon
nanopillars;
-
HF dip in order to etch away silicon dioxide mask leftovers;
-
two oxidation steps with HF etching in between in order to form Si nanocrystals inside
pillars.
4. 1. 2 Si nanopillars fabrication using negative resist
Next, in order to simplify the process, a negative resist was used, which hardens under the
exposure to the electron beam, in contrast to the described above positive type (Fig. 4.3):
- surface coating of the (001) Si chip with a negative resist (ma-N 2405);
- e-beam lithography to create dots with diameters 100-150 nm, resist developing;
- RIE in oxygen plasma to remove undeveloped resist around the structures and to clean area
around structures from remnants of the resist;
- dry etching of silicon in a RIE reactor using defined structures of negative resist as a mask.
- hydrogen peroxide etching to remove mask leftovers from the etched silicon structures.
- two oxidation steps with HF etching in between in order to form nanocrystals inside pillars;
Fig. 4.1 Electron beam lithography typically allows writing on the field size of 100x100 µm without
stage movements [2].
31
Chapter 4 Experimental methods___________________________________________________________
b)
a)
c)
d)
e)
f)
g)
Fig. 4.2 SEM images of nanopillar manufacturing process in positive resist, taken at different stages: (a)
holes with diameter ~ 150-200 nm are defined in a positive type of resist deposited on the surface of a Si
wafer (cleaved); (b) a zoomed image shows an “undercut” formed by scattered from the resist-silicon
interface electrons; (c) metallic (Cr) dots are formed on the sample surface after metal deposition and a
consecutive resist lift-off (top view); (d) a zoomed image reveals that it is possible to create almost
perfectly round metallic dots by such a technique; (e) pillars of Si ~ 200 nm tall are made using reactive
ion etching with previously formed structures as a mask (tilt 45o view); (f) a zoomed image reveals that
etching goes also in a lateral way, making the top of a pillar less in size than its base; (g) schematic
representation of the pillar manufacturing process using positive resist.
32
I. Sychugov
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
b)
a)
Fig. 4.3 SEM images of nanopillars manufacturing process in negative resist: (a) a set of resist dots
on silicon surface after exposure and development, (inset) zoomed in dot with characteristic diameter
of about 100 nm, (b) etched Si pillar with resist mask leftover on top.
4.1.3 Preparation of a Si nanopillar for TEM imaging
In order to correlate measured optical properties of a Si nanocrystal with its structural
features, such as size, shape, degree of stress a combined PL study with TEM characterization
should be applied. Here, we used the FIB method for TEM preparation. This technique allows
targeting specific area on the chip by milling a foil thin enough for subsequent TEM
characterization. However, very rigid site-selectivity requirements, i.e. we aim at a particular
pillar containing the luminescent nanocrystal, together with a small pillar diameter of ~ 30-40
nm made standard FIB milling method inapplicable.
A special method was developed to image single Si nanocrystals embedded into SiO2 pillars
with TEM. The main difficulty with conventional technique was the screening of the
structures by a high-Z protection material, normally deposited on a sample to suppress
charging during FIB processing. Thus a two stage protection scheme for FIB was developed
in order to replace high-Z materials on the edges of the pillar of interest with a low-Z
substance, namely a carbon-based polymer (positive resist ZEP520) (Fig. 4.4).
An additional problem was the fact that once being covered with resist, the structures become
inaccessible by SEM built-in to the FIB machine. Since electron beam is absorbed in the
deposited resist layer on top of structures it becomes impossible to target a specific area on
the chip with FIB. Thus, first, the area around the pillar of interest was to be precisely cleaned
from the resist. For this purpose e-beam lithography with a two-stage alignment routine was
used. The overall processing sequence is described below:
-
creating a hand-made mark on a chip with nanopillars of interest;
-
measuring coordinates of the nanostructures and hand-made marks on a chip with
SEM;
-
spinning ~ 500 nm thick positive resist to cover the structures with a low-Z material;
33
Chapter 4 Experimental methods___________________________________________________________
Fig. 4.4 Positive resist acting as a second protection layer in the preparation of nanopillars for TEM
with FIB processing. (Left) Optical microscope image of an array of pillars with three pillars
containing luminescent quantum dots. Area around the rows of interest is cleaned from the resist by
e-beam exposure in order to make it accessible in SEM. (Right) SEM image of a foil sputtered with
FIB; a two-layer protection is seen: the top, bright, layer is Pt deposited in FIB machine all over the
chip and the dark, middle layer is the positive resist covering the row of interest.
-
manually cleaning area on the edge of the sample where the hand-made mark is
located;
-
recalculating positions of the structures, using now-visible in SEM hand-made mark;
-
sending the beam to the newly calculated positions of the small alignment marks;
-
exposing 20×20 µm areas on them (typical positioning error due to uncertainty of the
stage movement ~ 10 µm);
-
developing resist thus cleaning small alignment marks from the resist;
-
locating now-visible small marks in SEM and performing fine alignment within one
writing field (typical positioning error < 100 nm);
-
exposing final patterns around the row with the pillar of interest;
-
developing resist to finalize resist openings around the row with the pillar of interest;
-
depositing second protective layer, Pt, over the whole chip in FIB machine;
-
digging out a section containing the pillar of interest with FIB (Fig. 4.4, right);
-
picking up the cut-out foil with a glass needle outside FIB machine;
-
placing it onto a standard grid for subsequent TEM observation or placing it onto a
SiO2 grid and subsequent oxygen plasma etching of the protective resist layer before
TEM observation.
34
I. Sychugov
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
Fig. 4.5 (Left) Negative resist dots formed by e-beam lithography with subsequent size-reduction by
reactive ion etching in oxygen plasma. The diameter is about 30 nm and the resist thickness ~ 200
nm. (Right) Silicon pillars with silica nanomesas doped with nanocrystals on top. Thus a point-like
light source is removed from the high refractive index substrate. The RIE etch did, however, not
work successfully in formation of vertical pillars and the diameter of Si pillar became larger then the
diameter of SiO2 nanomesas.
4.1.4 Formation of nano-mesas in SiO2 doped with Si nanocrystals
There are two main approaches in manufacturing of nanocrystals: top-down and bottom-up.
The first method, when a large structure is reduced to a nanometer-range object, is described
in previous sections. In this work another technique, when nanocrystals are formed by
precipitation of atoms, was also tested. Here, nanocrystals were formed by creating excess of
Si atoms by implantation in thermally grown silica and subsequent annealing. Atoms
precipitate around segregation centers and, when parameters of the annealing are chosen
properly, luminescent nanocrystals can be fabricated. The goal of this experiment was to etch
very small mesas in a thin layer of silica, doped with nanocrystals, in order to get access to a
single nanocrystal. This goal was never realized, however, possibly due to the still large
concentration of nanocrystals in thus-prepared mesas. The general process sequence was:
-
thermal growth of ~ 20 nm of SiO2 on top of a (001) Si wafer;
-
deposition of a sacrificial ~ 160 nm thick p-Si layer by the rf magnetron discharge;
-
introducing excess Si atoms (5-10 at. %) in SiO2 by implantation;
-
annealing at 1100 oC for 1 hour to form nanocrystals in the ~20 nm thin SiO2 layer;
-
wet etching of p-Si layer remnants;
-
spinning negative resist ~ 200 nm thick on top of a sample with an HMDS layer;
-
defining dots by e-beam lithography with diameter 100-200 nm;
-
RIE in oxygen plasma to reduce the size of the resist dots (down to ~ 30 nm);
-
transferring of the pattern to the SiO2 layer by RIE of the dioxide;
-
formation of Si pillars by RIE etching of Si (~ 250 nm) in order to distance light
emitters from the substrate;
35
Chapter 4 Experimental methods___________________________________________________________
a)
b)
c)
d)
Fig. 4.6 Preparation of tall pillars in silica doped with silicon nanocrystals: (a), (c) poly-Si pillars
acting as a mask for dioxide etching; (b), (d) final structures in silicon dioxide forming a photonic
crystal.
4.1.5 Formation of SiO2 nanopillars doped with Si nanocrystals
When tall pillars in silicon dioxide (~ 1 µm high) are to be fabricated, the negative resist
itself does not provide sufficient selectivity to SiO2 in RIE machine. At the same time its
thickness cannot be increased more than ~ 500 nm because resist pillars would not stand
upright due to the high aspect ratio. To circumvent this problem a double-mask approach was
used, when negative resist worked as mask for ~ 500 nm thick poly-Si layer, which, in turn,
acted as a mask for subsequent silicon dioxide etching. Here we targeted at the formation of a
photonic crystal structure composed of silicon dioxide pillars doped with silicon nanocrystals.
It was designed so that the bandgap matches the peak of the of luminescence spectra in order
to observe redistribution of light emission from silicon nanocrystals. This is an ongoing
project and the results are not published yet. The manufacturing procedure was as follows:
-
implantation of Si atoms into SiO2 substrate, thus introducing excess Si atoms in SiO2;
-
annealing at 1100 oC for 1 hour to form QDs in the top layer of ~ 1 µm thickness;
-
deposition of a ~ 500 nm poly-Si layer;
-
pattering the structure in the negative resist using e-beam;
-
transferring the pattern down to poly-Si by RIE (Figs 4.6 a, c);
-
transferring the pattern down to nc’s rich fused silica substrate by RIE (Figs 4.6 b, d).
36
I. Sychugov
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
Photoluminescence measurements
Luminescence from a semiconductor excited with photons (photoluminescence) or by electric
current (electroluminescence) is a powerful tool in studies of its properties. This method can
provide information on the optical bandgap, carrier relaxation, possible defect configurations
and a number of other characteristics (Fig. 4.7). A modified version of this technique – microPL – was used in the current work. The main advantage of it is the fact that not only spectral
characteristics of luminescence can be detected, but also the spatial configuration of it can be
mapped with resolution ~ 1 µm. The latter was used when the signal from a Si quantum dot
embedded in a silica nanopillar was to be addressed. When nanopillars are distanced more
than 1 µm from each other they can be separately probed by this method.
For PL characterization samples were mounted on a finger of the Oxford Instruments
Microstat cryostat. One can achieve temperatures of 5-10 K at the sample area with this
cryostat using liquid helium. However, a larger flux of the coolant usually brings significant
mechanical instability to the system, which is detrimental to long-term exposures.
Photoluminescence was excited by the 325 nm UV line of a He-Cd continuous wave laser IK
5351R-D by Kimmon Electric. The choice of excitation wavelength is based on the high
absorption cross-section of Si nanocrystals in this optical range. Another reason is the strong
absorption of UV light by the fused silica components of the microscope. This, in turn,
suppresses possible stray light influence on the signal. The emitted light is guided through the
microscope Nikon Optishot-150 (with objective lenses Nikon X100 for RT and windowcorrection Olympus X60 for low-T measurements) and through the imaging spectrometer
Jobin Yvon Triax 190 (Fig. 4.8). The latter has a turret with a mounted mirror and gratings of
different resolution. Depending on the task it is possible to monitor a white light reflection
image of the sample or, instead, acquire luminescent spectra from the area of interest. Finally,
the signal is collected by a Hamamatsu C4880 Dual-mode liquid nitrogen cooled CCD
camera. The sensitivity of about 3.8 photons per count appeared to be sufficient for single Si
Fig. 4.7 Schematic representation of the phenomenon of photoluminescence in bulk material; (a) an
electron-hole pair is created by an incident photon with energy exceeding the bandgap – nonresonant excitation; (b) by scattering with phonons carriers are thermalized to the edge of the
corresponding band; (c-e) various ways of carrier recombination with emission of a characteristic
photon: (c) band-to-band, (d) via a donor state, (e) via an acceptor state. A surface/interface state
can act as a donor or acceptor state in such a representation.
37
Chapter 4 Experimental methods___________________________________________________________
a)
b)
Fig. 4.8 The scheme of the experimental setup for PL studies. A sample is mounted on a finger of a
liquid helium cooled cryostat. Luminescence is excited by the UV-line (325 nm) of a cw He-Cd laser.
The emitted light is collected via an optical microscope through an imaging spectrometer onto a Si
chip-based CCD camera: (a) overall view of the setup [3]; (b) enlarged drawing of the sample area.
38
I. Sychugov
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
nanocrystal spectroscopy. Time-resolved PL measurements of ensembles of nanocrystals
were performed using a pulsed YAG laser at 355 nm with 3 ns pulse and 90 Hz repetition rate
and the photoluminescence emission was detected during a 10 µs-wide gate.
When a sample is cooled a condensate is formed on its surface. This frozen water and organic
compounds from the vacuum give significant bluish emission under UV excitation. The
spectral tail of it can interfere with the weak signal from nanocrystals, which is normally in
the red part of the visible spectrum. In order to avoid this effect a rapid heating of the cryostat
finger tip, where a sample is mounted, was carried out. The condensed vapors become volatile
around 273 K and get trapped on still cold parts of the cryostat, away from the tip. At this
moment heating is stopped and the sample is cooled again to desired cryogenic temperatures
yet without a luminescent film on the surface.
References
1. H. I. Liu, N. I. Maluf, R. F. W. Pease, D. K. Biegelsen, N. M. Johnson and F. A.
Ponce, J. Vac. Sci. Technol. B 10, 2846 (1992).
2. Manual for Raith 150 System, ver. 3.00.
3. J. Valenta, J. Linnros, R. Juhasz, F. Cichos, J. Martin in “Towards the First Silicon
Laser”, p. 89, Kluwer Academic Publishers (2003).
39
Chapter 5 Main results_____________________________________________________________________
Chapter 5. Main results
__________________________________________________
5.1 Optical properties of a single silicon nanocrystal
5.1.1 Photoluminescence spectroscopy
Silicon nanocrystals for single dot spectroscopy were formed inside silicon pillars by
oxidation, as shown by TEM micrographs in Figs. 5.1 (a)-(c). It is seen that the oxidation can
lead to the formation of a quantum dot near the top of the pillar (a) or a nanorod in the middle
of the pillar (b). Quite often, however, the pillar is fully oxidized and no silicon core is left
(c). This process is somewhat random and depends largely on the initial pillar size. Indeed,
the reduction from initially defined pillars of ~ 100 nm diameter down to ~ 5 nm is huge and
limits the precision of the remaining core. The effect of “self-limiting oxidation” is crucial in
formation of the quantum dot (see chapter 4). Note that images in Fig. 5.1 (a-c) reflect
ongoing work and these pillars do not contain a luminescent silicon nanocrystal.
d)
Fig. 5.1 (a)-(c) TEM micrographs of silicon pillars, which were oxidized at 9000C. The dark core is
remaining crystalline silicon, while bright shell around is amorphous SiO2. At certain oxidation stages
(a) or (b) a luminescent silicon nanocrystal with size <5 nm can be formed; (d) Array of pillars seen in
optical microscope and its corresponding photoluminescence image. Each bright spot can be traced
back to a certain pillar, which is indexed in the array.
40
I. Sychugov
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
Fig. 5.2 (a) Typical PL spectra from a Si nanocrystal at room temperature and at 80K. The featureless
spectrum at room temperature decomposes into the main line and the TO-phonon assisted one at 80
K (b) The ratio of quasidirect transition to the conventional silicon TO-phonon assisted one as a
function of emission photon energy: open circles for spherical quantum dots (ensemble
measurements from [2]), dark squares for samples from this work. Confinement energy is an addition
to the Si bandgap of ~ 1.1 eV, originating from the quantum confinement effect.
A large separation between neighboring pillars, as defined by e-beam lithography, made
possible far-field optics to be applied for characterization of these nanostructures. In Fig. 5.1
(d) a white light reflection image of the pillar array is shown together with a corresponding
PL image. Each light source can be traced back to a certain pillar indexed in the array.
In Fig. 5.2 (a) the spectra of a Si nanocrystal measured at room temperature and at 80 K are
shown. It is seen that a broad, featureless room-temperature spectrum decomposes into two
distinct contributions at lower temperature: the main line and the Si TO-phonon (~60 meV)
replica. The main line presumably originates from a quasidirect transition in silicon
Fig. 5.3 (a) Two spectra from different nanocrystals measured at 35 K. It is seen that the main line from
the previous figure is composed of a true main line and a ~6 meV sideband, tentatively attributed to
torsion or spherical acoustic phonon modes. (b) Statistical breakdown of the number of nanocrystals
with various emission photon energies. Altogether they fit into a typical ensemble spectra, measured
elsewhere (see, for example, [7]).
41
Chapter 5 Main results_____________________________________________________________________
nanocrystals, since if it were a conventional TO-phonon recombination the second peak
would correspond to the 2TO-phonon transition, which is highly unlikely from the
momentum conservation rule under non-resonant excitation conditions. Such quasidirect
transitions in silicon quantum dots were predicted [1] and indirectly observed [2] in ensemble
studies previously with the rate of the transition being a function of nanocrystal size.
Fig. 5.2 (b) summarizes the influence of emission photon energy (nanocrystal size) on the
probability of a quasidirect transition. The open circles are data obtained in [2] for spherical
nanocrystals and dark squares are values measured in this work. The clearly seen sizedependence for spherical nanocrystals can be readily explained considering larger degree of
electron and hole wavefunction overlap for a smaller nanocrystal [1]. The lack of a clearly
observed effect of confinement energy on the peak ratio in this work is ascribed to strong nonuniformity of nanocrystals prepared by the current method. Variations in shape or in the
degree of stress can enhance or decrease the probability of quasidirect transition in addition to
the frequently referred size effect. It was shown that an elongated nanorod of CdSe
nanocrystals [3], for example, has stronger oscillator strength of carrier interaction due to the
more pronounced effect of dielectric confinement. As for the effect of strain, theoretical
calculations [4] predicted that the existence of strained interface regions in the oxidized Si
nanocrystals, which is the case for the present experiment, leads to the localization of carriers
and a drastic increase of the no-phonon transition probability. In fact, only 1/3 of all measured
dots at 80 K revealed a TO-phonon replica, which suggests the dominating role of stress on
recombination in such prepared silicon nanocrystals.
Fig. 5.3 (a) shows how the emission line of a Si nanocrystal reduces further at a lowering of
the temperature down to 35 K. The fine structure of the main line is shown in Fig. 5.3 (a) for
two different nanocrystals (TO-phonon assisted lines are not shown). It is seen that this line
consists of a true main line, corresponding to the interlevel transition in nanocrystals, and a ~6
meV sideband, attributed tentatively to torsion or spherical acoustic phonon modes, predicted
[5] and indirectly observed [6] previously. The linewidth of the true main line is less than kBT
at this temperature, proving a discrete nature of energy states in a silicon quantum dot. At
even lower temperatures, however, no consistent PL was detected. We ascribe this effect to
singlet-triplet splitting of the first excited state in a silicon quantum dot, where a transition
from the lower lying triplet state to the ground level is optically forbidden. Thus only the
transition involving the singlet state can be detected and spectrally resolved by our system.
Hence, the PL intensity becomes too low to be detected by our system at temperatures below
~ 35 K.
Another important result of the work is presented in Fig. 5.3 (b), where it can be seen that
different nanocrystals possess various emission wavelengths presumably due to size
dispersion in accordance with the prediction of the quantum confinement theory. It is
important to note here that if the luminescence originated from some kind of defect or
interface state at this wavelength range, it would be impossible to observe such practically
continuous tunability of emission wavelength since each defect state would be a certain
configuration of Si/O atoms yielding a fixed energy of the corresponding optical transition.
42
I. Sychugov
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
5.1.2. Blinking phenomenon
The main results of the luminescence behavior investigation for Si nanocrystals in the time
domain are shown in Fig. 5.4. An example of the intensity trace is given in Fig. 5.4 (a) with
15 seconds integration time per interval. A reversible switching is seen between two discrete
states “on” and “off”.
The influence of the variable excitation power density on the blinking parameters was studied
and results are given in Fig. 5.4 (b). The underlying data analysis is outlined in Appendix A.
The switching rate for the on-off transition is clearly dependent on the pumping power, while
the reverse off-on process is nearly excitation independent. On the same figure the value of
the luminescence intensity in on-state is shown. The deviation from the linear regime and
bending of the curve towards the saturation takes place at the same excitation level as the
growth of the on-off switching rate. Such saturation-like behavior in the optical response of a
nanocrystal is typical when two excitons are formed at a time in a quantum dot. This effect
can be explained invoking a non-radiative Auger process, when energy from exciton
recombination is transferred to other available free carriers instead of being coupled to the farfield.
It would be reasonable to suggest a similar mechanism for the observed sensitivity of the onoff switching rate to the pumping power. Indeed, the energy obtained by another electron-hole
pair can be sufficient to move one of the carriers through or above the potential barrier
surrounding the nanocrystal to a localized trap state in silica bandgap. This effect charges the
nanocrystal and subsequent luminescence is quenched by three-particle Auger recombination.
A consequent restoring of the initial conditions by transfer of the carrier back to the
nanocrystal neutralizes an excess carrier and brings luminescence back in the on-state.
Fig. 5.4 (a) Luminescence behavior of a Si quantum dot in time domain. Sudden interruptions of light
emission – blinking – are seen. (b) Dependence of the blinking parameters on excitation conditions.
While the on-off switching rate grows quadratically with excitation power density, the reverse, off-on
switching frequency is basically independent on the pumping power.
43
Chapter 5 Main results_____________________________________________________________________
5.2 Effect of optical mode density on luminescence parameters
The influence of the high-n substrate proximity on optical properties of a quantum dot was
investigated. It is schematically shown in Fig. 5.5 (a) and (b) how the presence of a high
refractive index material can introduce anisotropy in the dipole emission. The two cases of
close proximity (a) and complete remoteness (b) were experimentally realized by distancing a
sub-wavelength light source, an ensemble of Si nanocrystals, away from the substrate using
reactive ion etching. Structures shown in Fig. 5.5 (c) and (d) correspond then to cases of
dipole emission depicted in Fig. 5.5 (a) and (b).
In Fig. 5.5 (e) the measured dependence of the total luminescence yield on remoteness from
the substrate for such a point-like light source is shown. The light was detected in the upper
hemisphere since the substrate material is opaque for these wavelengths (λ ≈ 800 nm here). It
is seen that at height ~ 250 nm a decoupling of the emitter from the substrate occurs and the
yield changes. Note that the pillar itself does not contribute much to the experiment, since
emission is only sensitive to the local optical environment in the sphere with a radius ~ λ/π.
The observed decoupling distance agrees well with that calculated in [8] for the depth of
perturbation ~ λ/π, caused by a high-n material on the interface dipole emission rate.
At the same time, along with the spatial redistribution of the light emission, the presence of a
high optical mode density substrate shortens the excitation lifetime of silicon nanocrystals.
However, real nanocrystals have QE<100%, which leads to the screeening of the changes in
the radiative rate for an observer measuring the total decay rate by the non-radiative
recombination inside the nanocrystals. From this approach one can roughly estimate the value
of quantum efficiency for nanocrystals. By measuring the change in the total decay rate with
decoupling emitter from the substrate and comparing to expected values for the radiative
component we estimate a QE ≈ 10 % for thus prepared silicon nanocrystals.
e)
Fig. 5.5 (a) A dipole emission pattern in proximity to a high refractive index substrate. (b) Isotropic
emission for a dipole suspended in air/vacuum. (c) Experimental realization of the case a): a pointlike light source, an ensemble of nanocrystals, is located directly on a high-n Si substrate. (d) An
ensemble of nanocrystal removed from the substrate by etching of Si, bringing the structure closer to
the case b) of dipole emission. (e) Dependence of the measured luminescence yield on remoteness of
the light source from a high optical mode density substrate (silicon, n ≈ 3.5).
44
I. Sychugov
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
References
1. M. Hybertsen, Phys. Rev. Lett. 72, 1514 (1994).
2. D. Kovalev, H. Heckler, M. Ben-Chorin, G. Polisski, M. Schwartzkopff and F. Koch, Phys.
Rev. Lett. 81, 2803 (1998).
3. A. Shabaev and Al. L. Efros, Nano Lett. 4, 1821 (2004).
4. C. Delerue, G. Allan and M. Lannoo, Phys. Rev. B 64, 193402 (2001).
5. T. Takagahara, J. Lumin. 70, 129 (1996).
6. M. Fujii, Y. Kanzawa, S. Hayashi, K. Yamamoto, Phys. Rev. B 54, R8373 (1996).
7. J. Linnros, N. Lalic, A. Galeckas and V. Grivickas, J. Appl. Phys. 86, 6128 (1999).
8. E. Snoeks, A. Lagendijk, and A. Polman, Phys. Rev. Lett. 74, 2459 (1995).
45
Chapter 6 Summary_______________________________________________________________________
Chapter 6. Summary
__________________________________________________
6.1 Comments on the appended papers
Paper I “Narrow luminescence linewidth of a silicon quantum dot”
The paper extends our previous work (Paper IV), where first results on low temperature
photoluminescence spectroscopy of single Si nanocrystals were reported. Here, the series of
experiments was performed at lower, cryogenic temperatures in order to get better insight into
the elementary processes of light emission from single silicon nanocrystals. This allowed to
observe that the structure of the main emission line consists of the sharp peak and a somewhat
broader satellite, separated by ~6 meV. We tentatively attributed this peak to a low-frequency
phonon assisted transition.
The main result of the paper, namely the broadening of the luminescence lines being less than
thermal in a certain temperature range, manifests that Si nanocrystals can be considered in a
similar way as their direct bandgap semiconductor counterparts, despite the substantial
difference in the zone structure of the corresponding bulk materials.
Paper II “Luminescence blinking of a Si quantum dot in a SiO2 shell”
In this work we turned to one of the most intriguing phenomena observed for quantum dots –
sudden luminescence intermittence. This phenomenon, often referred to as “blinking”, is
interesting not only from a physical viewpoint, but also has a certain practical value. Indeed,
the blinking lowers the effective quantum efficiency for a quantum dot and degrades the
performance of any device built on ensembles of quantum dots. Finally, the observation of
blinking can provide firm evidence that only a single light emitting center is probed. In order
to understand the physics behind the observed emission intermittence, sufficient statistics
should be acquired and data must be analyzed. The proper way of treating the recorded
intensity traces was not obvious from the very beginning. First, it was found that only traces
recorded with certain time resolution could be used in data analysis. It is only them, which
reveal blinking between two discrete levels (on/off). Since the switching rate is a function of
excitation intensity, the time resolution value is subject to changes under variable
experimental conditions. In other words, the proper time resolution must be chosen
individually for each set of experimental conditions.
As a result of such an analysis, we have found strong dependence for the on-off switching rate
on excitation power density. We ascribe this fact to Auger-assisted ionization, which
46
I. Sychugov
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
promotes charge fluctuation of a quantum dot. Moreover, SiO2 as a capping layer appeared to
be effective in charge capture, which manifests in various characteristic blinking times for
different nanocrystals, ascribed to variations in trap geometry around each dot.
Paper III “Effect of substrate proximity on luminescence yield from Si nanocrystals”
Here we investigate how the main parameters of nanocrystal luminescence, such as
spontaneous emission rate, quantum efficiency and luminescence yield are affected by
changes in its optical environment. Using e-beam lithography and reactive ion etching we
prepared a set of samples with point-like light sources (ensemble of nanocrystals) remote by
various distances from a high refractive index substrate (bulk Si). Small islands of
nanocrystals were located on top of pillars of various heights, representing point-like emitters
differently distanced from the substrate.
We applied both time-resolved and time-integrated methods to characterize the structures.
Time-resolved measurements were quite difficult due to the low emission rate from small
ensembles of nanocrystals. The time-resolved measurements yielded a higher decay rate
(shorter lifetime) when nanocrystals were in close proximity with the substrate. This would,
indeed, be anticipated due to the much higher density of optical modes in the substrate.
The results highlight the anisotropic nature of light emission at the interface. We found that
the luminescence yield can be increased by a factor of two by decoupling the light source
from the substrate. At the same time, in close proximity the quantum efficiency of silicon
nanocrystal doubles due to the leaking of light to the material with high density of optical
modes.
Paper IV “Single dot optical spectroscopy of silicon nanocrystals: low temperature
measurements”
The paper is a part of the Proceedings of the European Materials Research Society 2004,
Symposium A1: “Si-based photonics: towards true monolithic integration”. We included our
first results on low temperature photoluminescence spectroscopy of single Si nanocrystals.
There were a number of difficulties in the way of the experiment we report. The first and the
foremost is a very low emission rate of photons from a single silicon nanocrystal (of the order
of 10 detected photons per sec). Secondly, it was quite challenging to produce nanocrystals
well separated spatially so that far-field spectroscopy could be applied. Finally, all these
adversities were multiplied by the need for low-temperature measurements, where thermal
expansion of the cryostat finger must be suppressed to less than ~ 1 µm.
Here, we show that carefully prepared and patiently probed silicon nanocrystals do exhibit
relatively narrow luminescence linewidth at 80 K. In addition, the Si TO-phonon (~60 meV)
replica is observed for a fraction of investigated dots. Also we show that the temperature
bandgap narrowing, common for bulk Si, can be seen in our nanocrystals as well, which
supports the origin of light emission being that of band-to-band recombination.
Paper V “Light emission from silicon nanocrystals: probing a single quantum dot”
This paper was a contribution to the ACSIN-8 (Atomically Controlled Surfaces and
Interfaces) conference, which took place in the summer of 2005 in Stockholm. Here we
47
Chapter 6 Summary_______________________________________________________________________
analyzed data obtained by low-T spectroscopy of single silicon nanocrystals. The main
emphasis was made on the differences in some optical properties, such as linewidth and
luminescence yield, which we attributed to strong non-uniformity of quantum dots.
From this perspective it is important to combine optical characterization methods with
structural imaging in order to get information on both optical and morphological properties of
a single nanocrystal. This can help in understanding how size, stress or shape affects the
optical response from silicon nanocrystals.
Paper VI “Structural and optical properties of a single silicon quantum dot: towards
combined PL and TEM characterization”
In this work, which was a part of EMRS-2006 meeting, we report first results on combined
optical and structural characterization of single Si quantum dots. A special sample protection
scheme was developed in order to use a conventional focused ion beam machine for sample
preparation. The requirement of site-selectivity along with small size of pillars, containing the
quantum dots, made standard sample preparation techniques virtually impossible.
Using the e-beam machine we covered a row of pillars, containing the pillar of interest, with
resist, which is a carbon-based polymer. Afterwards the sample underwent Pt deposition to
suppress charging during the sputtering. Such a “sandwich” protection method allowed to
observe the silicon core in the oxidized pillars and to visualize luminescent silicon
nanostructures. One can see how the oxidation develops with time and at which stage
nanostructures appropriate for luminescence can be formed. To this point, we have not yet,
however, successfully correlated structural and optical properties on a single dot level.
6.2 Conclusions and outlook
The optical and electronic properties of semiconductors may change dramatically when scaled
into nanometer dimensions. Among semiconductors, silicon is the most ubiquitous electronic
material today and there is a chance this material may find new lease of life in the field of
optoelectronics in the future. Silicon nanocrystals (structures of less than 5 nm in size) are one
of several possible ways to obtain light emission from silicon. This is due to better
confinement of carriers, preventing them from migration to non-radiative centers, as occurs in
bulk material. Visible photoluminescence from silicon nanostructures (in the form of porous
silicon) was first observed more than a decade ago. However, the mechanism of that process
has remained the subject of debate for a long time. This is due to common observations of
very broad luminescence bands from practically all kinds of silicon nanoparticles.
In our project we designed and produced silicon nanocrystals in a silicon dioxide shell for
optical characterization by the low-temperature single-dot spectroscopy technique. This
method addresses one nanocrystal at a time and, avoiding inhomogeneous effects, represents
an effective tool for the investigation of basic properties of nanocrystals. Despite the obvious
advantages of this method, there were no reports on the application of such a technique to Si
nanocrystals. Although diluted porous Si particles were probed before, they still remain
conglomerates of several chromophores even after many filtering and thinning out steps.
On the experimental stage we indeed encountered many difficulties both in preparation and
characterization steps (Paper IV). One of the major physical obstacles was the long lifetime
48
I. Sychugov
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
(tens of microseconds) of the excitation in silicon nanocrystals, which still preserves the
indirect bandgap structure of bulk Si. This fact implies the need of a phonon for the
recombination of carriers, making it a low-probability three-particle process. Experimentally,
that corresponds to a very low signal-to-noise ratio when such nanostructures are optically
probed.
Being eventually successful, the experiments revealed several new physical properties of a
silicon quantum dot in a dioxide shell. For the first time narrow luminescence lines were
detected at low temperatures, corresponding to discrete energy levels of a Si nanocrystal
(Paper I). These lines were shifted in energy for different dots as would be expected from
differently sized nanocrystals. This result is a very strong indication towards quantum
confinement origin of light emission from silicon nanocrystals.
Luminescence blinking, i.e. arbitrary intermittence of emission was measured and analyzed
(Paper II). The bimodal distribution of luminescence intensity supports the origin of
luminescence being of a single emitter. In addition we found strong variation in luminescence
properties from dot-to-dot, ascribed to structural non-uniformity (Paper V). The latter
highlighted the need for both luminescence and structural characterization of such
nanostructures (Paper VI).
Our results, however, indicate major problems for the practical application of such
nanocrystals. Among them is the weak quasidirect recombination of carriers due to spatial
confinement of corresponding wave functions. We found that the no-phonon line can be only
several times stronger than the conventional, weak phonon-assisted one. In addition, another
disadvantage was confirmed by the blinking measurements. Namely, the high probability of
non-radiative Auger processes, which takes place when more than one electron-hole pair is
present in a Si quantum dot.
Possible ways to circumvent related difficulties would be using amorphous nanocrystals in
order to break crystalline order, or embedding them in, for example, a silicon nitride matrix.
In that case one can expect a dominance of quasidirect transitions in small nanocrystals.
Another approach is to modify their spontaneous emission rate (Paper III). In this work we
considered the effect of optical properties of the surrounding media on nanocrystal
luminescence. Quantum efficiency and luminescence yield were altered due to the
modification of the density of optical modes around nanocrystals and it was shown that the
value of quantum efficiency of these structures remains relatively high (~ 10 %). The
spontaneous emission rate could also be enhanced by coupling to surface plasmons on
nanostructured metals in proximity, which work as small antennas for coupling light to the
far-field.
This work gave rise to some intriguing questions, which appeared only after a single Si
nanocrystal had been accessed. Strong non-uniformity in observed nanocrystal optical
properties suggests the crucial role of the quantum dot shape, interface conditions and the host
matrix configuration on luminescence parameters. This is also true for local environment
around the nanocrystal. Indeed, observed variations in blinking suggest various scenario of
possible carrier capture in the surrounding shell for different nanocrystals.
To summarize, we investigated basic optical properties of silicon nanocrystals in a SiO2 shell
on a single-dot level, and probed into possible ways of modifying them. In our work for the
first time successful experimental probing of a single silicon nanocrystal was achieved.
49
Appendix A Blinking data analysis__________________________________________________________
Appendix A Blinking data analysis
__________________________________________________
Threshold approach in the analysis of the recorded PL traces
The most straightforward way of blinking analysis is based on a pre-estimated threshold
value, separating on and off states. Making frequency count (statistical analysis) of the
intensity trace, one can distinguish between these two states. Ideally, such statistical
breakdown of the intensity trace would yield two δ-functions, corresponding to each state. In
practice, the chosen time resolution, the Poisson noise of the detected counts and the CCD
read-out noise transform the δ-functions into Gaussian-shaped curves (Figs. A.1 (a) and (b)).
Such statistical analysis can reveal the presence of two discrete states only if the time
resolution is chosen correctly. If the chosen time interval value is too long, then many
intermediate states would enter and statistical breakdown of the trace would not reveal any on
and off states at all. On the other hand, the interval value should not be too small either due to
signal/noise considerations. These two limitations define the acceptable range for acquisition
time per frame. When the data were acquired with proper time resolution, a threshold value
can be established (Figs. A.1 (a) and (b)).
The statistics of the plateau durations for on and off states can be calculated, assuming that all
data points below the threshold correspond to the off state, while the rest – to the on state. In
Fig. A.2 calculated distributions are shown for two different values of excitation intensities
for the same quantum dot at RT. The fitting is based on a random switching model:
a)
b)
Fig. A.1 Statistical breakdown of intensity traces for dot A3IB at RT at two different excitation
intensities: a) 0.38 W/cm2 and b) 0.8 W/cm2. On and off states can be discerned and threshold values
can be estimated. Note that time-per-frame value is different in a) and b).
50
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
I. Sychugov
a)
b)
Fig. A.2 On and off times distributions for dot A3IB at RT at two different excitation intensities: a)
0.38 W/cm2 and b) 0.8 W/cm2, obtained from the threshold approach. The solid line is a fitting based
on the random switching model. Extracted values of τon and τoff are in time interval units ∆t.
⎛ t
p on ( t ) = A on ⋅ exp⎜⎜ −
⎝ τ on
⎛ t
p off ( t ) = A off ⋅ exp⎜⎜ −
⎝ τoff
⎞
⎟⎟
⎠
⎞
⎟⎟
⎠
(A.1a),
(A.1b).
where Aon,off – amplitudes, τon and τoff – average times in the on and in the off state,
respectively.
Thus two main parameters characterizing the blinking phenomenon τon and τoff can be found.
The main advantage of this method is its simplicity. However, there is some uncertainty in the
threshold value estimation due to the imperfectness of statistical distributions. This leads to a
slight error in determining τon and τoff.
Autocorrelation function
Along with the threshold approach for analyzing the PL trace of a quantum dot there is
another method based on the autocorrelation function. By definition, a second order
normalized autocorrelation function for a signal intensity trace I (t) is given by:
g ( 2) (τ) =
< I ( t ) > ⋅ < I ( t + τ) >
< I( t ) > 2
(A.2).
According to [3], for a random telegraph signal, whose on and off periods succeed one
another without any memory of former on and off times, the Laplace transform is related to
those of on and off time distributions:
~
~
1 ⎛ < τ off > ⎞ ⎛⎜
(1 − P) ⋅ (1 − Q) ⎞⎟
( 2)
~
⎟
⎜
⋅ 1−
g (s) = ⋅ ⎜1 +
~~
s ⎝
< τ on > ⎟⎠ ⎜⎝ s⋅ < τ on > ⋅(1 − PQ) ⎟⎠
51
(A.3),
Appendix A Blinking data analysis__________________________________________________________
~
~
where < τ on > ,( < τoff > ) - average on time (off time), P (s), Q (s) – the Laplace transforms of
the normalized on and off time distributions.
In [3] two particular cases were considered: the first when P (t) and Q (t) are power-law
dependencies and the second with P (t) – exponential and Q (t) – power-law. For the case of
silicon nanocrystals observed in this work, none of the models used in [3] are applicable.
Distributions of both on and off times, obtained from the threshold approach, appear to be
single-exponential (Fig. A.2). Therefore, an analytical expression for the autocorrelation
function should be calculated.
We use normalized single-exponential distributions: P(t) = (1/τon)exp(-t/τon), Q(t) =
(1/τoff)exp(-t/τoff) and their Laplace transforms:
∞
1
1 + s ⋅ τ on
(A.4a),
1
1 + s ⋅ τ off
(A.4b).
(s ⋅ τ off + 1)(τ on + τ off )
u (s)
~
≡
g ( 2 ) (s) =
s ⋅ (s ⋅ τ on ⋅ τ off + τ on + τ off ) v(s)
(A.5).
~
P(s) = ∫ P( x ) exp(−s ⋅ x )dx =
0
∞
~
Q(s) = ∫ Q( x ) exp(−s ⋅ x )dx =
0
Then (A.3) becomes
Using the Heaviside formula for the inverse Laplace transform for the case of polynomials
u(s) and v(s) in the numerator and denominator, respectively:
g ( 2) (τ) = ∑
k
a)
u (s k )
⋅ exp(s k ⋅ t ) ,
v | (s k )
b)
Fig. A.3 Calculated autocorrelation functions for PL traces for dot A3IB at RT at two different
excitation intensities: a) 0.38 W/cm2 and b) 0.8 W/cm2. Solid lines are fittings based on a
completely random switching model. Extracted values of τon and τoff are in interval units.
52
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
I. Sychugov
where sk – zeros of v polynomial, we obtain:
g ( 2) (τ) = 1 +
⎛ ⎛ 1
τ off
1
⋅ exp⎜⎜ − ⎜⎜
+
τ on
⎝ ⎝ τ on τ off
⎞ ⎞
⎟⎟ ⋅ τ ⎟
⎟
⎠ ⎠
(A.6).
By fitting this formula with two parameters τon and τoff to the autocorrelation function
calculated from the intensity trace (A.2), one can get values of τon and τoff (Figs. A.3 (a) and
(b)).
The advantage of using this approach is that no pre-defined threshold value is needed.
However, it can be seen in Fig. A.3 (b) that the fitting is to be done to one data point only
when the switching frequency is relatively high. The corresponding large error imposes
limitations on employing the autocorrelation function at this regime of blinking. On the other
hand, obtained frequency values for the lower switching regime are in good agreement with
those from the threshold approach (cf. Fig. A.2 (a)).
Switching frequencies for on-off and off-on processes
In the following chapter the formula for switching frequency of on-off and off-on processes
will be obtained from known τon and τoff values. Using the single-exponential distribution
(A.1a) one can write the total number of on-off switching events as
∞
N onoff = ∑ A on ⋅ exp(− n / τ on )
(A.7).
n =1
Total number of intervals spent in on-state:
∞
N on − state = ∑ A on ⋅ exp(− n / τon ) ⋅ n
(A.8).
n =1
Then, the switching frequency for on-off process
∞
f 10
N onoff 1
≡
⋅
=
N on −state ∆t
∑ exp(−n / τ
n =1
∞
∑ exp(−n / τ
n =1
on
on
)
)⋅n
⋅
1
∆t
(A.9),
or, using series sum formulas:
⎡
⎛ 1
f 10 = ⎢1 − exp⎜⎜ −
⎝ τ on
⎣
53
⎞⎤ 1
⎟⎟⎥ ⋅
⎠⎦ ∆t
[Hz]
(A.10a).
Appendix A Blinking data analysis__________________________________________________________
In a similar way, for off-on process:
⎡
⎛ 1
f 01 = ⎢1 − exp⎜⎜ −
⎝ τ off
⎣
⎞⎤ 1
⎟⎟⎥ ⋅
⎠⎦ ∆t
[Hz]
(A.10b),
where τon and τoff – average times in on and off state, [number of time intervals], ∆t – time
interval length, [seconds].
For large values of τon and τoff formulas (A.10a, b) can be simplified:
f 10 ≈
1 1
⋅ ,
τ on ∆t
f 01 ≈
1
τ off
⋅
1
[Hz].
∆t
These expression were used in [4] as “characteristic switching frequencies”.
Fraction of time spent in on-state
Another important parameter of the blinking process for a nanocrystal is the fraction of time
spent in on-state, αON. It can also be deduced from known τon and τoff values.
Using (A.8) one directly obtains
∞
α ON
N on −state
≡
=
N on −state + N off −state
∑A
n =1
∞
∑A
n =1
on
⋅ exp(−n / τ on ) ⋅ n
∞
on
⋅ exp(−n / τ on ) ⋅ n + ∑ A off ⋅ exp(− n / τ off ) ⋅ n
(A.11).
n =1
Taking into account the fact that the number of on-off switching events equals to the number
of off intervals (and vice versa):
∞
∞
n =1
n =1
∑ A on ⋅ exp(−n / τ on ) = ∑ A off ⋅ exp(−n / τ off )
(A.12),
equation (A.11) then becomes
∞
∑ exp(−n / τ
α ON =
n =1
on
)⋅n
,
∞
∞
∑ exp(−n / τ on ) ⋅ n +
n =1
∑ exp(−n / τ
n =1
∞
∑ exp(−n / τ
n =1
or, using series sum formulas:
54
on
off
)
)
∞
∑ exp(−n / τ
n =1
off
)⋅n
Synthesis and Properties of Single Luminescent Silicon Quantum Dots
I. Sychugov
α ON
⎛ 1 ⎞
⎟⎟ − 1
exp⎜⎜ −
τ
⎝ off ⎠
=
⎛ 1 ⎞
⎛ 1
⎟⎟ − 1 + exp⎜⎜ −
exp⎜⎜ −
⎝ τ off ⎠
⎝ τ on
⎞
⎟⎟ − 1
⎠
⋅ 100 ≡
f 01
⋅ 100 [%]
f 01 + f10
(A.13).
Again, in the case of large τon and τoff values, expansion in Taylor series yields a more simple
expression:
α ON ≈
τ on
⋅ 100 [%]
τ on + τ off
(A.14).
References
1. Al. L. Efros, M. Rosen, Phys. Rev. Lett. 78, 1110 (1997).
2. M. Kuno, D. P. Fromm, S. T. Johnson, A. Gallagher, D. J. Nesbitt, Phys. Rev. B 67,
125304 (2003)
3. R. Verberk, A. M. van Oijen, M. Orrit, Phys. Rev. B 66, 233202 (2002).
4. M.-E. Pistol, P. Castrillo, D. Hessman, J. A. Prieto, L. Samuelson, Phys. Rev. B 59, 10725
(1999).
55